A082695 -id:A082695 - cp's OEIS frontend

A385562 Numbers m such that (1/m) * Sum_{k=1..m} k/phi(k) sets a record value, where phi is the Euler totient function (A000010).

1, 2, 4, 6, 12, 18, 22, 24, 30, 42, 60, 66, 72, 78, 84, 90, 114, 120, 150, 156, 180, 198, 210, 300, 330, 390, 420, 510, 546, 570, 600, 630, 750, 780, 840, 966, 990, 1122, 1170, 1200, 1260, 1410, 1470, 1560, 1596, 1620, 1650, 1680, 1806, 1830, 1890, 1980, 2100

Offset: 1

Amiram Eldar, Jul 03 2025

nonn

Limit_{m->oo} (1/m) * Sum_{k=1..m} k/phi(k) = zeta(2)*zeta(3)/zeta(6) (A082695) (Sitaramachandrarao, 1985; Sándor et al., 2005). This sequence is infinite if this mean converges to the limit only from below.

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 29.

Amiram Eldar, Table of n, a(n) for n = 1..1752 (terms below 10^10)
R. Sitaramachandrarao, On an error term of Landau. II, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp. 579-588.

Cf. A000010, A068885, A069947, A082695, A330899, A385561.

seq[lim_] := Module[{s = {}, sum = 0, rm = 0, r}, Do[sum += k/EulerPhi[k]; r = sum/k; If[r > rm, rm = r; AppendTo[s, k]], {k, 1, lim}]; s]; seq[2500]

list(lim) = {my(sm = 0, rm = 0, r); for(k = 1, lim, sm += k/eulerphi(k); r = sm/k; if(r > rm, rm = r; print1(k, ", ")));}

A082696 Continued fraction expansion of Product_{p prime} (1+1/(p*(p-1))).

Original entry on oeis.org

1, 1, 16, 1, 2, 1, 2, 3, 1, 1, 3, 2, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 33, 2, 14, 1, 1, 1, 1, 1, 6, 2, 1, 3, 1, 3, 5, 3, 2, 13, 21, 1, 1, 1, 1, 8, 3, 27, 1, 2, 5, 1, 6, 2, 4, 3, 1, 1, 3, 10, 3, 5, 12, 48, 1, 2, 2, 1, 1, 1, 2, 2, 1, 11, 1, 1, 3, 7, 1, 30, 42, 1, 1, 1, 12, 1, 7, 1, 41, 1, 1, 1, 3, 5, 2, 1, 2, 7

Offset: 0

Benoit Cloitre, Apr 12 2003

Cf. A014197, A070243, A082695 (decimal expansion).

contfrac(prodeulerrat(1+1/(p*(p-1)))) \\ Amiram Eldar, Jun 13 2021

Offset changed by Andrew Howroyd, Jul 06 2024

A146323 a(n) = floor(Sum_{i=1..n} (1/phi(i))).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9

Offset: 1

Ctibor O. Zizka, Oct 30 2008

nonn

Looking on the number of 1's, 2's, ..., k's in this sequence we obtain the sequence (1,2,4,5,9,16,25,42,72,...). Limit_{k->oo} (number of (k+1)'s / number of(k's)) = sqrt(e).

The limit above is wrong. The correct limit is exp(zeta(6)/(zeta(2)*zeta(3))) = exp(1/A082695) = 1.672818789624... . - Amiram Eldar, Jul 04 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A028415, A048049, A074467, A082695.

IntegerPart[Accumulate[1/EulerPhi[Range[110]]]] (* Harvey P. Dale, Dec 19 2015 *)

list(lim) = {my(s = 0); for(k = 1, lim, s += 1/eulerphi(k); print1(floor(s), ", "));} \\ Amiram Eldar, Jul 04 2025

a(n) = floor(A028415(n)/A048049(n)). - Amiram Eldar, Jul 04 2025

A098468 Decimal expansion of constant A*B in the asymptotic expression of the summatory function Sum_{n=1..N} (1/phi(n)) as A(log(N)+B) + O(log(N)/N).

Original entry on oeis.org

0, 6, 0, 5, 7, 4, 2, 2, 9, 4, 8, 6, 3, 0, 5, 7, 3, 2, 1, 6, 0, 9, 7, 4, 4, 0, 1, 1, 6, 6, 3, 1, 3, 8, 4, 0, 3, 5, 4, 9, 7, 2, 2, 8, 4, 0, 8, 8, 2, 9, 8, 9, 2, 8, 1, 1, 5, 1, 2, 2, 4, 4, 8, 5, 6, 0, 9, 3, 4, 9, 8, 5, 5, 9, 0, 1, 8, 6, 4, 9, 1, 3, 1, 2, 3, 9, 2, 9, 8, 1, 5

Offset: 0

Eric W. Weisstein, Sep 09 2004

B equals EulerGamma - A085609.

			B = -0.0605742294.../A, where A is A082695.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 116.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (Z1*(gamma-Z2)).
Eric Weisstein's World of Mathematics, Totient Summatory Function

Cf. A082695, A085609.

(* Using S. Finch's notation *)
digits = 102;
A = Zeta[2]*Zeta[3]/Zeta[6];
S = Sum[Switch[Mod[k, 6], 0, 1, 1, 0, 2, -1, 3, -1, 4, 0, 5, 1]*PrimeZetaP'[k], {k, 2, 400}] // N[#, digits+40]&;
B = EulerGamma - S;
AB = A*B;
Join[{0}, RealDigits[AB, 10, digits][[1]]] (* Jean-François Alcover, Apr 28 2018 *)

Sum_{n=1..N} 1/phi(n) = A*(log(N)+B) + O(log(N)/N). - Jean-François Alcover, Apr 28 2018

More digits with the aid of A085609 and A082695 from R. J. Mathar, Jul 28 2010

More digits with the aid of A085609 and A082695 from Vaclav Kotesovec, Feb 17 2015

A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

Original entry on oeis.org

1, 0, 5, 6, 9, 3, 2, 2, 9, 1, 4

Offset: 1

Artur Jasinski, Jan 11 2021

Lesser twin primes A001359 (with the exception of the first prime, 3) are congruent to 5 mod 6: this constant is smaller than A340576.
By extrapolating method most probably the next two decimal digits are 1.056932291(46).
The known high-precision algorithms for Euler products are based on the Dirichlet L function and the Moebius inversion formula (see Mathematica procedure of Jean-François Alcover in A175646).
The constant is between 1.056932291453... and 1.056932291494. - R. J. Mathar, Feb 14 2025

			1.0569322914...

Cf. A005596, A065463, A065465, A065483, A065485, A065489, A065493, A072691, A082695, A111003, A175646.

One more digit confirmed by a bracketing of partial products - R. J. Mathar, Feb 14 2025

A373702 Decimal expansion of (2 - zeta(2))zeta(2)zeta(3)/zeta(6).

Original entry on oeis.org

6, 9, 0, 1, 0, 4, 8, 8, 2, 5, 1, 0, 2, 2, 4, 9, 7, 8, 1, 8, 7, 7, 3, 0, 0, 2, 5, 6, 7, 8, 2, 7, 5, 3, 2, 6, 4, 4, 0, 6, 6, 6, 2, 3, 1, 3, 1, 3, 3, 4, 8, 1, 2, 5, 4, 9, 1, 2, 2, 2, 9, 4, 2, 6, 0, 2, 0, 9, 9, 0, 1, 7, 1, 6, 8, 7, 3, 3, 7, 4, 6, 7, 2, 7, 9, 2, 6, 7, 8, 9, 1, 5, 0, 4, 0, 0, 5, 2, 5, 2

Offset: 0

Stefano Spezia, Jun 13 2024

			0.69010488251022497818773002567827532644066623131...

Steve Fan, The shifted prime-divisor function over shifted primes, arXiv:2406.05217 [math.NT], 2024. See p. 34.

Cf. A002117, A013661, A013664, A082695, A152416, A157289, A157292, A183699.

RealDigits[(2-Zeta[2])Zeta[2]Zeta[3]/Zeta[6],10,100][[1]]

cp's OEIS Frontend

A385562 Numbers m such that (1/m) * Sum_{k=1..m} k/phi(k) sets a record value, where phi is the Euler totient function (A000010).

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A082696 Continued fraction expansion of Product_{p prime} (1+1/(p*(p-1))).

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A146323 a(n) = floor(Sum_{i=1..n} (1/phi(i))).

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A098468 Decimal expansion of constant A*B in the asymptotic expression of the summatory function Sum_{n=1..N} (1/phi(n)) as A(log(N)+B) + O(log(N)/N).

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A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

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A373702 Decimal expansion of (2 - zeta(2))*zeta(2)*zeta(3)/zeta(6).

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A373702 Decimal expansion of (2 - zeta(2))zeta(2)zeta(3)/zeta(6).